Yours is an extremely important question!
As you pointed out, John 3:5 reads:
“[Truly], truly, I say to you, unless one is born of water [baptism] and the Spirit [immersion in the Word] he cannot enter into the kingdom of God.” (The bracketed notation is to clarify that baptism and revelation by the Spirit constitute our spiritual rebirth.)
Here, it is worthy of note that during physical birth, the fetus grows in physical, impure, amniotic fluid for about 40 weeks, which is analogous to our spiritual birth during baptism -- God does the cleansing. These same ideas are depicted in:
[Christ] loved the church... so that He might sanctify her, having cleansed her by [both] the washing of water with the word. The two clauses, the 1) washing of water is baptism and 2) with the Word is the revelation of the Holy Spirit.
We can see examples of this as Noah and his family was "baptized" in the Flood, one that occurred over 40 days and nights (again, remember that we grow physically in fluid for ~40 weeks). As long as they remained inside the ark [the church, their faithfulness], they were protected from the outside world of sin and death, one being cleansed by God with water. Too, the Israelites were "baptized" through the Red Sea (1 Cor. 10:2) after they fled Egypt (Egypt was a symbol of sin), where the Egyptian Army was washed away in the waters.
Further, had Naaman (2 Kgs. 5:14, Lk. 4:27) not obeyed the prophet Elisha's directive to immerse himself 7 times in the Jordan River (the "river of death"), his leprosy (another symbol of sin) would never have been washed away. Naaman's emergence from the water was a "rebirth." A thorough investigation of the symbolism of baptism throughout the Bible will reveal other examples that are quite intriguing.
However, some may fail to recognize such foundational truths. It was the universal practice in the early church that the new convert was baptized immediately. Contrary to what is sometimes taught, baptism is not merely “an outward sign of an inner grace”; such a description never appears in the Bible. A thorough study of the Book of Acts reveals the absolute necessity of water immersion -- baptism, into Christ (Mk. 16:16, Jn. 3:5, Acts 2:38, Acts 8:12, 16, Acts 10:48, Acts 19:5, Acts 22:16, Rom. 6:3, 1 Cor. 6:11, Gal. 3:27, Eph. 5:26, Col. 2:12, Tit. 3:5, 1 Pet. 3:21, etc.). The concept of an unbaptized Christian is simply nowhere entertained in the New Testament.
When we are baptized, we are cleansed of all our past sins and are raised in “newness of life” (Rom. 6:4, 7:6) -- we have "washed our robes in the blood of the Lamb" (Rev. 22:14) and have become sanctified as priests of God (1 Peter 2:9). Thereafter, as long as we walk in the Light of Christ, we are continually cleansed of all sin (1 Jn. 1:7) and all unrighteousness (1 Jn. 1:9) so that we may be presented before God holy, blameless, and beyond reproach (Col. 1:21-22). The text of 1 Peter 2 could not be clearer. When we internalize the Word of God, we are internalizing the Spirit of God; it becomes integral to our cognitive being. We grow in Christ through study, learning, and faithful obedience.
There's a very compelling case to be made from the book of Revelation, something not often appreciated in that symbolic work. I've used bracketed notation to paraphrase:
Revelation 20:6: “Blessed and holy is the one who has a part in the first resurrection [baptism into Christ]; over these [Christians], the second death [spiritual death] has no power, but they [Christians] will be priests of God and of Christ [cf. 1 Peter 2:9] and will reign with Him for a thousand years [an indeterminate period: the end of our lives].
Note the last part of the passage that states: “they will be priests of God and of Christ.” We read virtually identical language from Peter’s first epistle (to follow). Although it often seems difficult for us to fathom, given the many, fiery trials we often face, we are reigning with Christ in His Kingdom now (Matt. 23:14, 13:37-42, 44-46, 25:34, 41, Acts 28:30-31, Rom. 14:17, Col. 1:13, 4:11, Heb. 12:28, Rev. 11:15, etc.):
Here is the passage in question, regarding Rev. 20:6:
1 Peter 2:9-10: “[Christians] are a chosen race, a royal priesthood, a holy nation, a people for God’s own possession so that you may proclaim the excellencies of Him who has called you out of darkness into His marvelous light; for you once were not a people, but now you are the people of God; you had not received mercy, but now you have received mercy.”
The kingdom of Christ is the church, that being portrayed in Peter’s letter, just as it is in the Book of Revelation (20:6). Everyone may become a citizen of the Kingdom by appropriating salvation in Christ through His command to be baptized and immersed in the Spirit (the Word of God).